In both parts, you've started your argument by assuming what you're trying to prove.

In (a), you're being asked to prove that $[a]_6 = [b]_6 \implies ([a]_2, [a]_3) = ([b]_2, [b]_3)$, but you start by assuming this is true in the first place! What you could instead do is start with the idea in your second line to get something like:

In part (b), you've got the same issue. You're being asked to prove $f$ is an isomorphism, but you start by assuming it's an isomorphism! It seems like you've done most of the work needed for the proof (computing $f([a]_6$ for each $a$, showing $f$ respects addition, though you are missing an argument to show it respects multiplication) but you've got your argument back to front. You might like to have another attempt, bearing all this in mind.